\[ t^{p} - 1 = (t-1) (t-\omega) (t-\omega^{2}) \cdots (t - \omega^{p-1}) \]
より,
\[ (t-\omega) (t-\omega^{2}) \cdots (t - \omega^{p-1}) = 1 + t + t^{2} + \cdots + t^{p-1}. \]
\( t = 1 \) を代入して,
\[ (1-\omega) (1-\omega^{2}) \cdots (1 - \omega^{p-1}) = p. \quad \Box \]